Multiple Linear Regression

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predict(trees.lm, newdata = new, interval = "confidence")fit lwr upr1 8.264937 5.77240 10.757472 21.731594 20.11110 23.352083 30.379205 26.90964 33.84877Prediction intervals are given by> predict(trees.lm, newdata = new, interval = "prediction")fit lwr upr1 8.264937 -0.06814444 16.598022 21.731594 13.61657775 29.846613 30.379205 21.70364103 39.05477As before, the interval type is decided by the interval argument and the default confidencelevel is 95% (which can be changed with the level argument).Using the trees data,1. A 95% confidence interval for the mean Volume of a tree of Girth 9.1 in and Height 69 ftis given by [5.8, 10.8]. so with 95% confidence the mean Volume lies somewhere between5.8 cubic feet and 10.8 cubic feet.2. A 95% prediction interval for the Volume of a hypothetical tree of Girth 12.5 in and Height87 ft is given by [26.9, 33.8], so with 95% confidence the hypothetical Volume of a tree ofGirth 12.5 in and Height 87 ft would lie somewhere between 26.9 cubic feet and 33.8 feet.3 Model Utility and Inference3.1 <strong>Multiple</strong> Coefficient of DeterminationThe error sum of squares S S E can be conveniently written in MLR asS S E = Y T (I − H)Y. (31)The ANOVA decomposition saysS S TO =Y T (I − 1 n J )Y (32)10
andS S R =Y T (H − 1 n J )Y. (33)Immediately from Equations 31, 32, and 33 we get the Anova EqualityS S TO = S S E + S S R. (34)The multiple coefficient of determination is defined by the formulaR 2 = 1 − S S ES S TO . (35)R 2 is the proportion of total variation that is explained by the multiple regression model. InMLR we must be careful, however, because the value of R 2 can be artificially inflated by theaddition of explanatory variables to the model, regardless of whether or not the added variables areuseful with respect to prediction of the response variable.We address the problem by penalizing R 2 when parameters are added to the model. The resultis an adjusted R 2 which we denote by R 2 .R 2 =(R 2 −p ) (n − 1n − 1n − p − 1). (36)It is good practice for the statistician to weigh both R 2 and R 2 during assessment of model utility.In many cases their values will be very close to each other. If their values differ substantially, orif one changes dramatically when an explanatory variable is added, then (s)he should take a closerlook at the explanatory variables in the model.3.2 How to do it with RFor the trees data, we can get R 2 and R 2 from the summary output or access the values directlyby name as shown (recall that we stored the summary object in treesumry).> treesumry$r.squared[1] 0.94795> treesumry$adj.r.squared[1] 0.9442322High values of R 2 and R 2 such as these indicate that the model fits very well, which agrees withwhat we saw in Figure 2.11
Page 5: We see from the output that for the
Page 12 and 13: 3.3 Overall F-TestAnother way to as
Page 14 and 15: ●Volume10 30 50 70● ●●●
Page 16 and 17: Coefficients:Estimate Std. Error t
Page 18 and 19: F-statistic: 350.5 on 2 and 28 DF,p
Page 21 and 22: Height by a new variable Tall which
Page 23 and 24: eforehand and wrote the discussion
Page 25 and 26: for this hypothesis test there are
Page 27 and 28: ---Signif. codes: 0 ‘***’ 0.001
Page 29 and 30: In the trees example we may take th
Page 31 and 32: Here is an example of how it works,

Multiple Linear Regression

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